In this paper, we prove some new inequality of Adams’ type for some new higher order Sobolev space whose norm is a combination of the norms of the \( \frac{N}{2}-\) Bilaplacian and the \( p-\) Bilaplacian with \( p < \frac{N}{2} \) in the whole euclidean space \( \mathbb {R}^N,\ N \ge 4. \) The inequality proved is completely new. Next, an improvement of this inequality, inspired by the concentration-compactness principle of P. Lions, is also provided. This improvement is not trivial and its proof needs some new sophisticated tools. Finally, using this inequality, we treat in the last part of this work, some biharmonic elliptic quasilinear equation involving \( (\frac{N}{2},p)-\) Bilaplacian operators and where the nonlinearities enjoy an exponential growth condition at infinity.
